You will have to install a domain controller with the DAG environment. Without it the Exchange platform performance will be very poor and if you lose your main office you would have no Exchange. As you

I would think you would need a VPN for the replication to work. You can do this on the same network as the MAPI connection, although best practice dictates that a dedicated replication network be used.

Latency might be an issue here, but it all depends on the connection speed. Do you have this information?

Are you looking to create a new AD site in the cloud and install a Domain Controller (GC) and mailbox server role etc? If so there are some cluster commands you should be aware of to do with timeouts when a DAG is stretched over a WAN.

You will have to install a domain controller with the DAG environment. Without it the Exchange platform performance will be very poor and if you lose your main office you would have no Exchange. As you have to use Enterprise edition, use virtual machines and have the DC in another VM.

One option would be to have just one member of the DAG in the cloud, along with a DC. Keep the active database local. That will reduce the bandwidth use.